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MATH 621 – Spring 2005 Homework Set # 9
- Let z 0 , z 1 , z 2 , z 3 be four points in the extended complex plane. Let ξ = [z 0 , z 1 , z 2 , z 3 ] be their cross-ratio.
a) Give a careful definition of the cross-ratio in case one of the points is the point at infinity.
b) Show that the cross-ratios of the 24 permutations of z 0 , z 1 , z 2 , z 3 may take only one of the following values:
ξ,
ξ
, 1 − ξ,
1 − ξ
ξ ξ − 1
ξ − 1 ξ
- Two points z 0 and z∗ 0 in the plane are said to be symmetric with respect to the circle or line defined by three distinct points z 1 , z 2 , z 3 if and only if: [z 0 , z 1 , z 2 , z 3 ] = [z∗ 0 , z 1 , z 2 , z 3 ].
a) Prove that a fractional linear transformation T maps a pair of points z 0 , z 0 ∗ , symmetric with respect to the line/circle defined by z 1 , z 2 , z 3 to a pair of points symmetric with respect to the line/circle defined by T (z 1 ), T (z 2 ), T (z 3 ).
b) Give a geometric interpretation of symmetry in the case when the three points z 1 , z 2 , z 3 lie on a line.
c) Give a geometric interpretation of symmetry in the case when the three points z 1 , z 2 , z 3 are not collinear.
- It is shown in the textbook (page 219) that if |α| < 1, the fractional linear transformation
T (z) =
z − α αz ¯ − 1 is an automorphism of the unit disk D and that it satisfies T 2 = 1. Let T be a fractional linear transformation mapping the unit disk D to itself. Show that T may be written as
T (z) = eiθ^
z − α αz ¯ − 1
where θ ∈ R and |α| < 1 Hint: Let α ∈ D be such that T (α) = 0. Show that T (1/ α¯) = ∞.
- Suppose the 2 × 2 matrix:
M =
a b c d
; ad − bc 6 = 0 c 6 = 0
has two distinct eigenvalues λ and λ′^ with |λ| > |λ′|. Let ( w 1 w 2
w′ 1 w′ 2
2
be the corresponding eigenvectors. Set w = w 1 /w 2 and w′^ = w′ 1 /w′ 2 Let TM : C → C be the associated fractional linear transformation.
a) Show that w and w′^ are distinct fixed points of TM. Conversely, show that if TM has two distinct fixed points in C, then M has two distinct eigenvalues.
b) Prove that for any z ∈ C,
lim k→∞
T (^) Mk (z) = w.
- Find (if impossible, explain why) a fractional linear transformation which:
a) Maps the circle {|z| = 1} to itself and maps 0 to 1/2.
b) Maps the circles {|z| = 1} and {|z| = 2} to parallel lines.
c) Maps the region inside the circle {|z| = 2} and outside the circle {|z + 1| = 1} to the region between the horizontal lines v = 0 and v = 1.
- Let f (z) be holomorphic in the open disk DR = {|z| < R}. Suppose that |f (z)| < M for all z ∈ DR and f (0) = 0.
a) Prove that |f (z)| ≤
M
R
|z| for all z ∈ DR, with equality holding at
some point different from the origin if and only if f (z) = eiβ^
M
R
z.
b) Suppose f has a zero of order k at 0. Prove that f (z) ≤
M
R
|z|k
for all z ∈ DR, with equality holding at some point different from the
origin if and only if f (z) = eiβ^
M
R
zk^.
- Let f (z) be holomorphic in the unit disk ∆ = {|z| < 1 }. Suppose that Im(f (z)) > 0 for all z ∈ ∆ and that f (0) = i. Show that for all z ∈ ∆:
|f (z)| ≤
1 + |z| 1 − |z|
- Let f : ∆ → ∆ be a holomorphic map from the unit disk into itself. Prove that for all a ∈ ∆ we have |f ′(a)| 1 − |f (a)|^2
1 − |a|^2
Hint: Let g be an automorphism of ∆ such that g(0) = a and let h be an automorphism of ∆ which maps f (a) to 0. Let F = h ◦ f ◦ g. Compute f ′(0) and apply the Schwarz Lemma.
- Let C∗^ = C \ { 0 }. Compute Aut(C∗).
